Mathematical Descriptions of The Electromagnetic Field - Geometric Algebra Formulations

Geometric Algebra Formulations

Analogous to the tensor formulation, two objects, one for the field and one for the current, are introduced. In geometric algebra (GA) these are multivectors. The field multivector, known as the Riemann–Silberstein vector, is

and the current multivector is

where, in the algebra of physical space (APS) with the vector basis . The unit pseudoscalar is (assuming an orthonormal basis). Orthonormal basis vectors share the algebra of the Pauli matrices, but are usually not equated with them. After defining the derivative

Maxwell's equations are reduced to a single equation,

Maxwell's equations (APS formulation)

In three dimensions, the derivative has a special structure allowing the introduction of a cross product:

from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation. After expanding and rearranging, this can be written as

\left( \boldsymbol{\nabla} \cdot \mathbf{E} - \frac{\rho}{\epsilon_0} \right)- c \left( \boldsymbol{\nabla} \times \mathbf{B} - \mu_0 \epsilon_0 \frac{\partial {\mathbf{E}}}{\partial {t}} - \mu_0 \mathbf{J} \right)+ i \left( \boldsymbol{\nabla} \times \mathbf{E} + \frac{\partial {\mathbf{B}}}{\partial {t}} \right)+ i c \left( \boldsymbol{\nabla} \cdot \mathbf{B} \right)= 0

We can upgrade from APS to spacetime algebra (STA) by reassigning with unit pseudoscalar . The new basis vectors share the algebra of the gamma matrices but like above are usually not equated with them. The derivative is now

It is customary to leave the Riemann–Silberstein vector as a bivector

but to reduce the current to a vector

Owing to the very simple identity

all of Maxwell's equations reduce to an even terser single equation:

Maxwell's equations (STA formulation)

Read more about this topic:  Mathematical Descriptions Of The Electromagnetic Field

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